On what days of week during the Gregorian period can a century begin?

similar to my code for another one of your problems
we can see that the only days of the week found are Monday, Tuesday, Thursday and Saturday

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

<?php
$results = array();
$days = array('sun','mon','tue','wed','thu','fri','sat');
for($i=0;$i<=6;$i++){
    $results[$days[$i]] = 0;
}
for($y=2001;$y<=2400;$y+=100){
    $time = mktime(0,0,0,1,1,$y);
    $results[$days[date('w',$time)]]++;
}
echo '<pre>'.print_r($results,true).'</pre>';
/*
Array
(
    [sun] => 0
    [mon] => 1
    [tue] => 1
    [wed] => 0
    [thu] => 1
    [fri] => 0
    [sat] => 1
)
*/
?>

January 1, 2001, was in fact a Monday.

An ordinary Gregorian century has 76 ordinary and 24 leap years. $76\times 365+24\times 366 \mod 7=5$ .

A leap Gregorian century has 75 ordinary and 25 leap years. $75\times 365+25\times 366 \mod 7=6$ .

Note, $3\times 5+6 \mod 7=0$ , which proves the four Gregorian century cycle repeats the day of week pattern.

Five days after Monday is Saturday. Five days after Saturday is Thursday. Five days after Thursday is Tuesday .